basis-map coefficients of A371080, shifted → basis A013610

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(-2 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	4	15	9													28
3	28	117	108	27												280
4	280	1254	1431	594	81											3640
5	3640	17142	22365	12015	2835	243										58240
6	58240	285192	409266	259335	81405	12393	729									1106560
7	1106560	5593368	8631630	6155163	2324700	479682	51030	2187								24344320
8	24344320	126373776	206675964	161308476	69608889	17527104	2561706	201204	6561							608608000
9	608608000	3232377360	5546020428	4652739792	2224147653	647004267	116623962	12715218	767637	19683						17041024000
10	17041024000	92332390080	164985704064	146914775460	76234353660	24788562435	5206483737	705897990	59639490	2854035	59049					528271744000
11	528271744000	2913427164480	5391553996224	5049315151452	2804009289840	997148496465	235766683152	37502288901	3966518160	267393555	10392624	177147				17961239296000
12	17961239296000	100641338824320	192053117365056	187851377138040	110484261308916	42315076749330	11007512716563	1982377872090	247368484143	20990935350	1155529881	37200870	531441			664565853952000
13	664565853952000	3777613254387840	7407889358980032	7526660306202648	4651471799844012	1897110623651958	534223200760821	106370519417019	15099767529561	1518770060379	105727411647	4843021833	131265927	1594323		26582634158080000
14	26582634158080000	153098227737369600	307648414122364800	323290080325046016	208638852912368424	89838840345610356	27060259901388714	5857490378963223	923102259433497	106050105003843	8785406647017	510903108261	19779702579	457570701	4782969	1143053268797440000

Computed in your browser ?

Technical details

Node id: n-b0592fccca597966 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f061f7c708547767

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A007559: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         1,          3;
  2:         4,         15,          9;
  3:        28,        117,        108,         27;
  4:       280,       1254,       1431,        594,         81;
  5:      3640,      17142,      22365,      12015,       2835,       243;
  6:     58240,     285192,     409266,     259335,      81405,     12393,       729;
  7:   1106560,    5593368,    8631630,    6155163,    2324700,    479682,     51030,     2187;
  8:  24344320,  126373776,  206675964,  161308476,   69608889,  17527104,   2561706,   201204,   6561;
  9: 608608000, 3232377360, 5546020428, 4652739792, 2224147653, 647004267, 116623962, 12715218, 767637, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n - 2 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 2 + 3*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8877bf17db51e7c6
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-8877bf17db51e7c6
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-8877bf17db51e7c6

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013610 e-8877bf17db51e7c6

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-aa6a1f197b91b3b6 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.